In complex and algebraic geometry, various notions of hyperbolicity have been introduced to study the geometry of projective manifolds of general type. In this talk, we will first introduce the notions of Brody hyperbolicity and algebraic hyperbolicity for smooth projective curves and surfaces, and discuss some classical results. We will then shift to the logarithmic analogue, introducing the algebraic exceptional set associated with the complement of a plane curve and discussing recent progress in this direction. N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).